Suppose given a holomorphic and Hamiltonian action of a compact torus T on a polarized Hodge manifold M. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of T on the associated Hardy space. If in addition the moment map is nowhere zero, for each weight ν the ν-th isotypical component in the Hardy space of the polarization is finite-dimensional. Assuming that the moment map is transverse to the ray through ν, we give a geometric interpretation of the isotypical components associated to the weights kν, k→+∞, in terms of certain polarized orbifolds associated to the Hamiltonian action and the weight. These orbifolds are generally not reductions of M in the usual sense, but arise rather as quotients of certain loci in the unit circle bundle of the polarization; this construction generalizes the one of weighted projective spaces as quotients of the unit sphere, viewed as the domain of the Hopf map.

Paoletti, R. (2021). Polarized orbifolds associated to quantized Hamiltonian torus actions. JOURNAL OF GEOMETRY AND PHYSICS, 170(December 2021) [10.1016/j.geomphys.2021.104363].

Polarized orbifolds associated to quantized Hamiltonian torus actions

Paoletti R.
2021

Abstract

Suppose given a holomorphic and Hamiltonian action of a compact torus T on a polarized Hodge manifold M. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of T on the associated Hardy space. If in addition the moment map is nowhere zero, for each weight ν the ν-th isotypical component in the Hardy space of the polarization is finite-dimensional. Assuming that the moment map is transverse to the ray through ν, we give a geometric interpretation of the isotypical components associated to the weights kν, k→+∞, in terms of certain polarized orbifolds associated to the Hamiltonian action and the weight. These orbifolds are generally not reductions of M in the usual sense, but arise rather as quotients of certain loci in the unit circle bundle of the polarization; this construction generalizes the one of weighted projective spaces as quotients of the unit sphere, viewed as the domain of the Hopf map.
Articolo in rivista - Articolo scientifico
Geometric quantization; Hamiltonian actions; Hardy space; Polarized orbifold; Unit circle bundle;
English
31-ago-2021
2021
170
December 2021
104363
partially_open
Paoletti, R. (2021). Polarized orbifolds associated to quantized Hamiltonian torus actions. JOURNAL OF GEOMETRY AND PHYSICS, 170(December 2021) [10.1016/j.geomphys.2021.104363].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/329318
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